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Problem 3. Let $n \ge 2$ be an integer. Euroland has $n$ cities, with direct flights connecting every pair of cities in both directions. For each pair of cities, the emperor assigns a positive price, which is the same in each direction. For two distinct cities $A$ and $B$, let $D(A, B)$ be the number of flights in the cheapest journey between them; if there are multiple such journeys, then $D(A, B)$ is defined by the longest one.
For each value of $n$, find the largest average value of $D(A, B)$ over all pairs of distinct cities $(A, B)$, that the emperor can achieve.
Solution 1Solution 2Solution 3Solution 4
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